Odd Legs

Absence of gap for infinite half-integer spin ladders with an odd number of legs.

A proof is presented for the absence of gap for spin 1/2 ladders with an odd number of legs, in the infinite leg length limit. This result is relevant to the current discussion of coupled one–dimensional spin systems, a physical realization of which are vanadyl pyrophosphate, (VO)2P2O7, and stoichiometric Srn−1 Cun+1 O2n (with n = 3, 5, 7, 9, . . .). The magnetic properties of low dimensional s...

Odd Independent Transversals are Odd

We put the final piece into a puzzle first introduced by Bollobás, Erdős and Szemerédi in 1975. For arbitrary positive integers n and r we determine the largest integer ∆ = ∆(r, n), for which any r-partite graph with partite sets of size n and of maximum degree less than ∆ has an independent transversal. This value was known for all even r. Here we determine the value for odd r and find that ∆(...

Skolem Odd Difference Mean Graphs

In this paper we define a new labeling called skolem odd difference mean labeling and investigate skolem odd difference meanness of some standard graphs. Let G = (V,E) be a graph with p vertices and q edges. G is said be skolem odd difference mean if there exists a function f : V (G) → {0, 1, 2, 3, . . . , p + 3q − 3} satisfying f is 1−1 and the induced map f : E(G) → {1, 3, 5, . . . , 2q−1} de...

Artificial Legs

The restless legs syndrome.

The first description of the restless legs syndrome (RLS) is attributed to Thomas Willis in the 'London Practice of Physick' of 1685 (Ekbom, 1960). Here he wrote 'wherefore to some, when being a bed they betake themselves to sleep, presently in the arms and leggs, leapings and contractions of their tendons, and so great a restlessness and tossings of their members ensue, that the diseased are n...